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Theorem isso2i 5032
Description: Deduce strict ordering from its properties. (Contributed by NM, 29-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
isso2i.1 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦𝑦𝑅𝑥)))
isso2i.2 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Assertion
Ref Expression
isso2i 𝑅 Or 𝐴
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧

Proof of Theorem isso2i
StepHypRef Expression
1 equid 1936 . . . . 5 𝑥 = 𝑥
21orci 405 . . . 4 (𝑥 = 𝑥𝑥𝑅𝑥)
3 eleq1 2686 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
43anbi2d 739 . . . . . 6 (𝑦 = 𝑥 → ((𝑥𝐴𝑦𝐴) ↔ (𝑥𝐴𝑥𝐴)))
5 equequ2 1950 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥 = 𝑦𝑥 = 𝑥))
6 breq1 4621 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦𝑅𝑥𝑥𝑅𝑥))
75, 6orbi12d 745 . . . . . . 7 (𝑦 = 𝑥 → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥 = 𝑥𝑥𝑅𝑥)))
8 breq2 4622 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
98notbid 308 . . . . . . 7 (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
107, 9bibi12d 335 . . . . . 6 (𝑦 = 𝑥 → (((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦) ↔ ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥)))
114, 10imbi12d 334 . . . . 5 (𝑦 = 𝑥 → (((𝑥𝐴𝑦𝐴) → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦)) ↔ ((𝑥𝐴𝑥𝐴) → ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))))
12 isso2i.1 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦𝑦𝑅𝑥)))
1312con2bid 344 . . . . 5 ((𝑥𝐴𝑦𝐴) → ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦))
1411, 13chvarv 2262 . . . 4 ((𝑥𝐴𝑥𝐴) → ((𝑥 = 𝑥𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))
152, 14mpbii 223 . . 3 ((𝑥𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
1615anidms 676 . 2 (𝑥𝐴 → ¬ 𝑥𝑅𝑥)
17 isso2i.2 . 2 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1813biimprd 238 . . . 4 ((𝑥𝐴𝑦𝐴) → (¬ 𝑥𝑅𝑦 → (𝑥 = 𝑦𝑦𝑅𝑥)))
1918orrd 393 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
20 3orass 1039 . . 3 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
2119, 20sylibr 224 . 2 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
2216, 17, 21issoi 5031 1 𝑅 Or 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3o 1035  w3a 1036  wcel 1987   class class class wbr 4618   Or wor 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-po 5000  df-so 5001
This theorem is referenced by:  ltsonq  9743  ltsosr  9867  ltso  10070  xrltso  11926
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